Integer partitions

A partition writes a positive integer as an unordered sum of positive integers.

Different ways to understand it

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Catalan numbers
Catalan numbers count several recursively structured objects, including balanced parenthesis strings.
Group actions
A group action assigns compatible transformations of a set to group elements.
Inclusion–exclusion
Alternating intersection counts correct double-counting in unions of finite sets.
Ramsey theory
Sufficiently large structures force specified regular substructures under suitable coloring conditions.
Binomial theorem
Binomial coefficients give the expansion of an integer nonnegative power of a sum.